super-d numbers

For

, a super-d number is a number $n$

such that $d\cdot n^d$ contains a substring made of $d$

digits

For example, 261 is a super-3 number since $3\cdot261^3=5\underline{333}8743$ .

The first super-d numbers are

d	super-d
2	19, 31, 69, 81, 105, 106, 107, 119, 127, 131,...
3	261, 462, 471, 481, 558, 753, 1036, 1046, 1471,...
4	1168, 4972, 7423, 7752, 8431, 10267, 11317,...
5	4602, 5517, 7539, 12955, 14555, 20137, 20379,...
6	27257, 272570, 302693, 323576, 364509, 502785,...
7	140997, 490996, 1184321, 1259609, 1409970,...
8	185423, 641519, 1551728, 1854230, 6415190,...
9	17546133, 32613656, 93568867, 107225764,...

The first super-d numbers are 19, 31, 69, 81, 105, 106, 107, 119, 127, 131, 169, 181, 190, 219, 231, 247, 261, 269, 281, 310, 318, 319, 331 more terms

Below, the spiral pattern of super-d numbers up to $250^2$ shows some long runs of consecutive terms. See the page on prime numbers for an explanation and links to similar pictures.

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many super-d numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Super-d Number
OEIS, Sequence A032743

Super-d numbers can also be... (you may click on names or numbers and on + to get more values)

a-pointer 181 631 3373 + 9978131 ABA 81 338 1058 + 9981512 aban 19 31 69 + 9000988 abundant 318 336 348 + 9999894 Achilles 3087 10584 17496 + 9984600 admirable 318 1036 3342 + 9999668 alt.fact. 19 619 326981 3301819 alternating 69 81 105 + 9898981 amenable 69 81 105 + 9999981 amicable 71145 142310 185368 + 9592504 apocalyptic 247 669 749 + 29981 arithmetic 19 31 69 + 9999981 astonishing 3388 7119 78403 178623 balanced p. 1123 4013 4597 + 9997219 bemirp 1061 1901 19001 + 1909081 betrothed 1050 549219 1057595 + 7890575 binomial 105 190 231 + 9916831 brilliant 169 247 319 + 9999557 c.decagonal 31 281 781 + 9975781 c.heptagonal 106 1072 1331 + 9990436 c.nonagonal 190 1081 2926 + 9916831 c.octagonal 81 169 1369 + 9979281 c.pentagonal 31 106 181 + 9965031 c.square 181 481 2381 + 9985981 c.triangular 19 31 631 + 9988471 cake 988 10701 19650 + 9002008 Canada 581 Carmichael 10585 172081 252601 + 9890881 Carol 261119 Chen 19 31 107 + 9999931 congruent 31 69 119 + 9999981 constructible 3084 3855 4369 + 8355840 cube 1331 3375 6859 + 9393931 Cullen 106497 229377 Cunningham 31 127 511 + 9966650 Curzon 69 81 105 + 9999981 cyclic 19 31 69 + 9999931 D-number 69 219 339 + 7042881 d-powerful 332 333 334 + 9995572 de Polignac 127 331 337 + 9999869 decagonal 2232 10660 14701 + 9955602 deceptive 481 2981 3367 + 9890881 deficient 19 31 69 + 9999981 dig.balanced 19 169 318 + 9999981 double fact. 105 135135 droll 33280 48384 161280 + 6848512 Duffinian 81 119 169 + 9999919 eban 2046 4056 30036 + 6066054 economical 19 31 81 + 9999931 emirp 31 107 337 + 9999931 emirpimes 169 319 335 + 9999919 enlightened 2097152 5117695 equidigital 19 31 81 + 9999931 eRAP 10620 26123 121539 + 9750699 esthetic 454 5454 7656 + 9876545 Eulerian 247 15619 16369 + 4194281 evil 105 106 119 + 9999981 fibodiv 19 969 3379 + 6384052 Fibonacci 4181 1346269 9227465 Friedman 127 1792 3281 + 995331 frugal 1331 1369 1681 + 9992403 gapful 105 190 231 + 9999810 Gilda 35422 3363582 4484776 good prime 127 269 331 + 9993019 happy 19 31 190 + 9999919 harmonic 8190 105664 237510 + 753480 Harshad 81 190 247 + 9999881 heptagonal 81 469 783 + 9957046 hex 19 127 169 + 9964519 hexagonal 190 231 1431 + 9916831 highly composite 15120 332640 498960 hoax 319 336 454 + 9999895 Hogben 31 381 601 + 9976123 Honaker 131 1049 2591 + 9985231 house 12131 66859 78338 + 9566817 hungry 82810 2401519 hyperperfect 10693 306181 808861 + 9699181 iban 107 127 247 + 777743 iccanobiF 1053 104937 1454698 idoneal 105 190 462 1848 impolite 16384 2097152 inconsummate 261 381 431 + 999931 interprime 69 81 105 + 9999895 Jacobsthal 2731 87381 Jordan-Polya 13824 16384 138240 + 4838400 junction 105 107 319 + 9999753 Kaprekar 181819 818181 katadrome 31 81 310 + 9876543 Kynea 1050623 Lehmer 247 481 511 + 9996205 Leyland 2169 533169 1048976 + 1596520 lonely 24281 31431 58831 370310 Lucas 15127 64079 1860498 + 7881196 lucky 31 69 105 + 9999807 Lynch-Bell 784 1764 1926 + 9867312 m-pointer 1123 1231 2131 + 2121131 magic 369 7825 19669 + 9410681 magnanimous 281 310 334 + 9171170 metadrome 19 69 127 + 2346789 modest 19 69 333 + 9998181 Moran 190 247 261 + 9999881 Motzkin 127 15511 nialpdrome 31 81 310 + 9999981 nonagonal 261 969 5226 + 9862446 nude 333 336 784 + 9999666 oban 19 69 310 + 988 octagonal 481 1281 4256 + 9977280 odious 19 31 69 + 9999895 Ormiston 1931 19013 19031 + 9994631 palindromic 131 181 333 + 9990999 palprime 131 181 919 + 9907099 pancake 106 631 781 + 9992686 panconsummate 31 69 81 + 481 pandigital 19 1054 1070 + 9998058 partition 231 12310 105558 + 7089500 pentagonal 247 715 782 + 9991051 pernicious 19 31 69 + 9999895 Perrin 119 2627 3480 + 3932465 Pierpont 19 769 10369 + 331777 plaindrome 19 69 119 + 7778899 Poulet 4369 4681 8481 + 9890881 power 81 169 784 + 9979281 powerful 81 169 784 + 9984600 practical 336 348 462 + 9999664 prim.abundant 318 1036 2470 + 9999668 prime 19 31 107 + 9999931 primeval 107 1012379 primorial 2310 9699690 pronic 462 1056 1190 + 9900462 Proth 81 481 769 + 9965569 pseudoperfect 318 336 348 + 999894 repdigit 333 3333 33333 + 3333333 repfigit 19 31331 34285 + 355419 repunit 31 127 381 + 9976123 Rhonda 8526 12985 33475 + 8511529 Ruth-Aaron 105 369 715 + 9825127 Saint-Exupery 33600 1054560 1063860 + 9903180 Sastry 715 106755 self 31 310 334 + 9999919 self-describing 444422 semiprime 69 106 119 + 9999919 sliding 15689 78253 156890 + 7825300 Smith 319 454 690 + 9999895 Sophie Germain 131 281 419 + 9998903 sphenic 105 190 231 + 9999869 square 81 169 784 + 9979281 star 181 337 2281 + 9930493 straight-line 333 369 531 + 9876543 strobogrammatic 69 181 619 + 9980866 strong prime 107 127 269 + 9999419 super Niven 1050 3330 10500 + 9300060 superabundant 15120 332640 tau 348 1056 1068 + 9999072 taxicab 134379 327763 515375 + 8493039 tetrahedral 969 9880 10660 + 9290431 tetranacci 10671 20569 1055026 triangular 105 190 231 + 9916831 tribonacci 81 10609 410744 trimorphic 749 781249 890624 truncatable prime 31 337 719 + 9973547 twin 19 31 107 + 9999931 uban 19 31 69 + 9000081 Ulam 69 106 131 + 9999569 undulating 131 181 454 + 9696969 unprimeable 1070 1072 1145 + 9999895 untouchable 336 782 784 + 999894 upside-down 19 753 1469 + 9955511 vampire 105210 105264 105750 + 809919 wasteful 69 190 231 + 9999981 weak prime 19 31 131 + 9999931 weakly prime 971767 1062599 1524181 + 9537371 weird 10570 11690 16310 + 985810 Wieferich 10533 42627 127881 Woodall 5119 106495 229375 + 9961471 Zeisel 105 1419 5719 + 9773731 Zuckerman 1131 2232 7119 + 9414144 Zumkeller 318 336 348 + 99894 zygodrome 333 1188 3322 + 9999666